In three dimensions, the Weyl tensor is identically zero and all information
about conformal geometry is encoded in the Cotton tensor, \(C_{abc}=- C_{bac}\) .
Spacetime is conformally flat if and only if the Cotton tensor vanishes [1]}
(see [2]} for a modern proof). The commutator of two conformally covariant derivatives \(\nabla _{a}\)  involves only the Cotton tensor [2]} (see also [4]})
\(\big [\nabla _a,\nabla _b\big ] =\frac{1}{2}C_{abc}K^c~, \qquad C_{abc}=-\varepsilon _{abd}C^d{}_{c}~,\) 
